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Problem. Find L[J0(t)] where J0(t) is the Bessel function of order zero.


Solution.


Method 1, using series. J0(t) is given by


             ole.gif


Applying the formula


             ole1.gif  


to each of the terms we get


             ole2.gif


                                     ole3.gif



Now it happens that the quantity in brackets is the binomial expansion of


 

             ole4.gif


Thus

 

             ole5.gif




Method 2, using differential equations. The function J0(t) satisfies Bessel’s differential equation which reduces in the case of J0(t) to


ole6.gif


We shall now take the Laplace transform of both sides of 1). Letting y = L [J0(t)] and using the theorems

ole7.gif


together with J0(0) = 1, J0'(0) = 0, we obtain


             ole8.gif


which gives


ole9.gif


Expanding and rearranging we obtain


ole10.gif


or


ole11.gif



We now integrate both sides to obtain


ole12.gif


or


ole13.gif


Thus


ole14.gif


where c is an arbitrary constant. We now determine the value of the constant c as follows: Multiply 7) by s and take the limit as s ole15.gif to give


ole16.gif


Now since


ole17.gif


we deduce by the initial-value theorem that c = 1. Thus we obtain

 

             ole18.gif


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